The average salary of all the workers in a workshop is Rs 8000. The average salary of 7 technicians is Rs 12000 and the average salary of the rest is Rs 6000. The total number of workers in the workshop is:

Options

A. 20
B. 21
C. 23
D. 22

Answer

To solve the problem step by step, we will use the concept of averages and the relationship between the total salaries and the number of workers.

### Step 1: Define the variables
Let:
- \( N \) = total number of workers in the workshop
- \( S \) = total salary of all workers
- \( S_T \) = total salary of technicians
- \( S_R \) = total salary of the rest of the workers

### Step 2: Calculate the total salary of all workers
Given that the average salary of all workers is Rs 8000, we can express the total salary \( S \) as:
\[ S = N \times 8000 \]

### Step 3: Calculate the total salary of technicians
We know there are 7 technicians with an average salary of Rs 12000. Therefore, the total salary of the technicians \( S_T \) is:
\[ S_T = 7 \times 12000 = 84000 \]

### Step 4: Calculate the total salary of the rest of the workers
Let \( R \) be the number of remaining workers. The average salary of the rest of the workers is Rs 6000. Thus, the total salary of the rest \( S_R \) can be expressed as:
\[ S_R = R \times 6000 \]

### Step 5: Relate the total number of workers
Since the total number of workers \( N \) is the sum of technicians and the rest, we have:
\[ N = 7 + R \]

### Step 6: Write the total salary equation
The total salary \( S \) can also be expressed as the sum of the salaries of technicians and the rest:
\[ S = S_T + S_R \]
Substituting the values we have:
\[ N \times 8000 = 84000 + R \times 6000 \]

### Step 7: Substitute \( R \) in terms of \( N \)
From the equation \( N = 7 + R \), we can express \( R \) as:
\[ R = N - 7 \]
Substituting this into the total salary equation gives:
\[ N \times 8000 = 84000 + (N - 7) \times 6000 \]

### Step 8: Expand and simplify the equation
Expanding the equation:
\[ N \times 8000 = 84000 + N \times 6000 - 42000 \]
Simplifying:
\[ N \times 8000 = N \times 6000 + 42000 \]

### Step 9: Rearranging the equation
Rearranging gives:
\[ N \times 8000 - N \times 6000 = 42000 \]
\[ N \times (8000 - 6000) = 42000 \]
\[ N \times 2000 = 42000 \]

### Step 10: Solve for \( N \)
Dividing both sides by 2000:
\[ N = \frac{42000}{2000} = 21 \]

### Conclusion
The total number of workers in the workshop is \( N = 21 \).

---